An imperceptible connection between the Clebsch--Gordan coefficients of $U_q({\mathfrak{sl}}_2)$ and the Terwilliger algebras of Grassmann graphs

Abstract

The Clebsch--Gordan coefficients of $U({\mathfrak{sl}}_2)$ are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism $♮$ from the universal Hahn algebra $\mathcal{H}$ into $U({\mathfrak{sl}}_2)\otimes U({\mathfrak{sl}}_2)$. Let $\Omega$ denote a finite set of size $D$ and $2^{\Omega }$ denote the power set of $\Omega$. It is generally known that ${\mathbb{C}}^{2^{\Omega }}$ supports a $U({\mathfrak{sl}}_2)$-module. Let $k$ denote an integer with $0\le k\le D$ and fix a $k$-element subset $x_0$ of $\Omega$. By identifying ${\mathbb{C}}^{2^{\Omega }}$ with ${\mathbb{C}}^{2^{\Omega \setminus x_0}}\otimes {\mathbb{C}}^{2^{x_0}}$ this induces a $U({\mathfrak{sl}}_2)\otimes U({\mathfrak{sl}}_2)$-module structure on ${\mathbb{C}}^{2^{\Omega }}$ denoted by ${\mathbb{C}}^{2^{\Omega }}(x_0)$. Pulling back via $♮$ the $U({\mathfrak{sl}}_2)\otimes U({\mathfrak{sl}}_2)$-module ${\mathbb{C}}^{2^{\Omega }}(x_0)$ forms an $\mathcal{H}$-module. When $1\le k\le D-1$ the $\mathcal{H}$-module ${\mathbb{C}}^{2^{\Omega }}(x_0)$ enfolds the Terwilliger algebra of the Johnson graph $J(D,k)$ with respect to $x_0$. This result connects these two seemingly irrelevant topics: The Clebsch--Gordan coefficients of $U({\mathfrak{sl}}_2)$ and the Terwilliger algebras of Johnson graphs. Unfortunately some steps break down in the $q$-analog case. By making detours, the imperceptible connection between the Clebsch--Gordan coefficients of $U_q({\mathfrak{sl}}_2)$ and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper.

arXiv comment65 pages

References in corpus (6)

• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• Entanglement of Free Fermions on Johnson Graphs
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
• The Clebsch--Gordan coefficients of $U({\mathfrak{sl}}_2)$ and the Terwilliger algebras of Johnson graphs
• Grassmann graphs, degenerate DAHA, and non-symmetric dual $q$-Hahn polynomials